Abstract
Under some conditions, an asymptotic solution containing boundary functions of two types was constructed by V.F. Butuzov and N.N. Nefedov for initial value problems for differential equations involving the second power of a small parameter multiplying the derivative with a right-hand side consisting of a singular matrix \(A(t)\) times the unknown function (as a linear part of the equation) plus the same small parameter multiplying a nonlinear function. In the present paper, an algorithm for constructing asymptotics using the orthogonal projectors onto \(\text{ker}A(t)\) and \(\text{ker}A(t)'\) (the prime denotes transposition) is given. This approach can be useful for understanding the algorithm underlying the construction of asymptotics. It allows us to present formulas for finding asymptotic terms of any order in an explicit form.
Similar content being viewed by others
REFERENCES
A. B. Vasil’eva and V. F. Butuzov, Singularly Perturbed Equations in the Critical Case (Mosk. Gos. Univ., Moscow, 1978; Univ. of Wisconsin-Madison, Madison, 1980).
R. E. O’Malley, Jr., “A singular singularly-perturbed linear boundary value problem,” SIAM J. Math. Anal. 10 (4), 695–708 (1979).
H. K. Khalil, “Feedback control of nonstandard singularly perturbed systems,” IEEE Trans. Autom. Control 34 (10), 1052–1060 (1989).
V. F. Butuzov and A. B. Vasil’eva, “Differential and difference systems of equations with a small parameter in the case when the unperturbed (degenerate) system is situated on the spectrum,” Differ. Equations 6, 499–510 (1971).
Z. M. Gu, N. N. Nefedov, and R. E. O’Malley, Jr., “On singular singularly perturbed initial value problems,” SIAM J. Appl. Math. 49 (1), 1–25 (1989).
C. Schmeiser and R. Weiss, “Asymptotic analysis of singular singularly perturbed boundary value problems,” SIAM J. Math. Anal. 17 (3), 560–579 (1986).
R. E. O’Malley, Jr., and J. E. Flaherty, “Analytical and numerical methods for nonlinear singular singularly-perturbed initial value problems,” SIAM J. Appl. Math. 38 (2), 225–248 (1980).
U. Ascher, “On some difference schemes for singular singularly-perturbed boundary value problems,” Numer. Math. 46, 1–30 (1985).
V. F. Butuzov and N. N. Nefedov, “A problem in singular perturbation theory,” Differ. Equations 12, 1219–1227 (1976).
A. B. Vasil’eva, V. F. Butuzov, and L. V. Kalachev, The Boundary Function Method for Singular Perturbation Problems (SIAM, Philadelphia, 1995).
G. A. Kurina, “Projector approach to constructing asymptotic solution of initial value problems for singularly perturbed systems in critical case,” Axioms 8 (2), 56 (2019). https://doi.org/10.3390/axioms8020056
G. A. Kurina and N. T. Hoai, “Projector approach for constructing the zero order asymptotic solution for the singularly perturbed linear-quadratic control problem in a critical case,” International Conference on Analysis and Applied Mathematics (ICAAM 2018), AIP Conf. Proc. 1997, 020073-1–020073-7 (2018).
G. Kurina and N. T. Hoai, “Zero order asymptotic solution of a class of singularly perturbed linear-quadratic problems with weak controls in a critical case,” Optim. Control Appl. Methods 40 (5), 859–879 (2019).
Y. Sibuya, “Some global properties of matrices of functions of one variable,” Math. Ann. 161, 67–77 (1965).
T. Kato, Perturbation Theory for Linear Operators (Springer-Verlag, Berlin, 1966).
O. Bretscher, Linear Algebra with Applications, 5th ed. (Pearson, Harlow, 2013).
Ju. L. Daleckiĭ and M. G. Kreĭn, Stability of Solutions of Differential Equations in Banach Space (Nauka, Moscow, 1970; Am. Math. Soc., Providence, 1974).
A. B. Vasil’eva and V. F. Butuzov, Asymptotic Expansions of Solutions of Singularly Perturbed Equations (Nauka, Moscow, 1973) [in Russian].
G. A. Kurina and N. T. Hoai, “Projector approach to constructing asymptotic solution of initial value problems for a class of singularly perturbed equations in a critical case,” Proceedings of the 12 International Conference on Management of Large-Scale System Development (MLSD'2019), Moscow, October 1–3, 2019 (Inst. Probl. Upr. Ross. Akad. Nauk, Moscow, 2019), pp. 981–984.
ACKNOWLEDGMENTS
We are deeply grateful to V.F. Butuzov for helpful discussion of this paper.
Funding
The work of G.A. Kurina was supported by the Russian Science Foundation, grant 17-11-01220. The work of N.T. Hoai was supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED), grant 101.02-2017.314.
Author information
Authors and Affiliations
Corresponding authors
Additional information
Dedicated to Professor V.F. Butuzov on the occasion of his 80th birthday
Rights and permissions
About this article
Cite this article
Kurina, G.A., Hoai, N.T. Projector Approach to the Butuzov–Nefedov Algorithm for Asymptotic Solution of a Class of Singularly Perturbed Problems in a Critical Case. Comput. Math. and Math. Phys. 60, 2007–2018 (2020). https://doi.org/10.1134/S0965542520120076
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0965542520120076